In Theorem REF  we provide a general criterion for the weak convergence of
discrete-time coalescent processes to \(\Xi \) -coalescents as \(N\rightarrow \infty \) ; which in particular applies to the
genealogy of GWF populations but also to (non-heritable) general offspring distributions.
This in fact follows by a careful study of the corresponding arguments
in [1]} for Cannings' populations.
Of course any vector of offspring sizes \((\xi _{1,N},\dots ,\xi _{N,N})\)  becomes exchangeable once we
randomly permute its entries, thus effectively rendering any model with non-heritability into a Cannings' model.
This loss of information on the asymmetric reproductive success of the population will, however, overcomplicate the
study of its genealogy. Theorem REF , and its extension for GWF populations provided by Theorem REF , allows us to
harness the asymmetry of \(\eta ^{(N)}\)  to more easily derive the limit genealogy of the population.
In Proposition REF , we show how Theorem REF  can exploit this asymmetry
when the limit genealogy is a \(\Lambda \) -coalescent, simplifying the proofs and generalizing similar results
found in the literature.
The same techniques can also be used for the general
\(\Xi \) -coalescent case; for simplicity and space constraints we do not include such development here.
